Assignment I Pareto Distribution

Transcription

1 Name and Surname: Thi Thanh Tam Vu PhD student in Economics and Management Course: Statistics and regressions ( ) Assignment I Pareto Distribution 1.! Definition The Pareto distribution (hereinafter as PD), named after the Italian civil engineer, economist and sociologist Vilfredo Pareto, is a power law 1 probability distribution that is used in description of social, scientific, geophysical, actuarial, and many other types of observable phenomena. Outside economics, it is also known as Bradfold distribution. There is the so-called Generalized Pareto Distribution (GPD), which is a family of continuous probability distributions. In GPD, here is a hierarchy of Pareto distributions known as Pareto Type I, II, III, IV, and Feller Pareto distributions. Within the scope of this report, Pareto distribution (type I) would be taken in account. Figure 1: An example of Pareto distribution 2.! Main properties If X is a random variable with a PD (type I), then P(X>x) =! # $%&$' ') 1$$$$$$$$%&$' < ' ) Where, ' ) is a positive minimum possible value of x, called a scale parameter and, is a positive parameter, called a shape parameter or tail index. Support: [' ), ] 1!In statistics, a power law is a functional relationship between two quantities: one quantity varies as a power of another. In other words, a relative change in one quantity would cause to a proportional relative change in other quantity regardless to the initial values of these quantities.!!! 1

2 Mode: ' ) Median: Cumulative distribution function (c.d.f) 0 1 ' = $ ' # ) 1 $ $%&$' ' ' ) $ 0$$$$$$$$$$$$$$$$%&$' ' ) Probability density function (p.d.f) & 1 ' = $ Expected value (mean),' ) # ' #67 $%&$' ' )$ 0$$$$$%&$' ' ) < < E(X) = '& ' 8' = ' # 9 8' =! 9:; # $,' 7 < ) 7=# '7=# Then, E(X) = $$$$$$$%&$, 1 # #=7 $$%&$, > 1 Variance With, > 1 < Var(X) = ' # Then, Var(X) = The k th moment #=7? #!" 9 8'! 9:; $$$$$$$$$$$$$$$ $$$$$$$$$$%&$, (1,2] # C (#=7) C (#=?) $$%&$, > 2 Figure 2: Prob. density and distribution functions of Pareto distribution for various, E(' E )=! F " $E! #=7 (#=E) (, > I; I$%K$LMK%N%OP$%QNPRPS) 3.! Applications The Pareto distribution is a highly left skewed distribution defined in terms of its scale and a shape factors. It is a heavy tailed distribution meaning that a random variable following a Pareto distribution can have large values, this makes it a tool for analyzing extreme! 2

3 values. Pareto distribution is first used by Pareto when he described the allocation of wealth as in any society, a small percentage of population inclines to hold a large proportion of properties of that society. Later on, he also used it in the description of the allocation of income. His work is then known as rule, meaning that 80% of the national wealth owns by only 20% of the population. After that, the Pareto distribution is manipulated to illustrate the distribution of the small to the large : 80% of something comes from 20% of the population. Later analysis of statistical distributions has demonstrated that Pareto distributions were indeed very common in various fields -! Enterprises distribution by employee size -! The sizes of human settlements (few cities, many hamlets/villages) -! The values of oil reserves in oil fields (a few large fields, many small fields) -! Severity of large casualty losses for certain lines of business such as general liability, commercial auto and worker compensation -! In hydrology, the Pareto distribution is applied to extreme events such as annually maximum one-day rainfalls and river discharges. In particular, according to Klugman, Panjer and Wilmot (2004), Pareto distribution has many economic applications. Since it is a heavy tailed distribution, it is a good candidate for modeling income above a theoretical value and the distribution of insurance claims above a threshold value. Pareto distribution is most well-known in characterizing the distribution of income (even wage/salary) and wealth, which are represented by the Lorenz curve. Figure 3 illustrates the Lorenz curve for a number of Pareto distribution.! 3

4 Figure 3: Lorenz curves for a number of Pareto distribution: T = corresponds to the perfectly equal distribution of income while T = U corresponds to the most unequal distribution Lorenz curve L(F) is defined in terms of p.d.f or c.d.f of Pareto by the following formula: Lorenz curve then provides the information about the inequality in income in a country. Moreover, the c.d.f of Pareto distribution also yields the GINI coefficients, which measures the family-income inequality in a country on the scale from zero (perfect equality) and 100 (the most unequal). GINI index is calculated as follows: In Management, Pareto distribution is widely used in a great deal of activities such as quality control, project management, reducing customer complaints, reducing organizational and system problem. In general, Pareto distribution would be used in business process and efficiency improvement. 4.! Relation to other distributions Generalized Pareto distribution Pareto distribution is a special case of the Generalized Pareto distribution with location V = ' ), scale W = # and shape X = 7 #.! 4

5 Exponential distribution -! If X is Pareto- distributed with scale parameter ' ) $and shape parameter,, then Y=log( 1 ) is exponentially distributed. Proof: F(y)= Pr(Y Y) = Pr$(log$( 1 ) Y) = Pr$( 1 P _ ) = Pr$(` ' ) P _ ) Since X is Pareto distributed, we have Pr(X ' ) P _ ) = 1 a b # = 1 P =#_ Hence, Y=log( 1 )$is exponentially distributed. -! Similarly, if r.v.y is exponentially distributed with parameter,>0, then r.v. X= ' ) P c is Pareto-distributed with min ' ) $and index,. Zeta distribution Zeta distribution can be considered as a discrete counterpart of Pareto distribution. In case of discrete probability distribution, if X is a zeta-distributed random variable with parameter s, then the probability that X takes the integer value k is given by the probability mass function, which is 5.! How to determine probabilities using a computer The probabilities of Pareto distribution could be determined by using several software specialized on statistical analysis. Apart from those tools, it is also convenient to use R- a programming language and software environment for statistical computing and graphics. In R, the probabilities of Pareto distribution can be illustrated through a set of the so-called R-type functions. Several R-type functions are listed in Figure 4. Nevertheless, before conducting those functions, it is necessary to download a package called VGAM by the following function: And then, open the library VGAM by the subsequent function:! 5

6 Figure 4: Standard R-type functions for determining probabilities of Pareto distribution Notice that since there are four types of Pareto distribution, the number which stands for the type of Pareto should be clearly indicated. For example, we could compute the probability density and probability distribution of Pareto distribution type I with a vector of quantiles q and the shape and scale parameter are 2 and 3 respectively as follows: References Downey, A. B. (2005). Lognormal and Pareto distributions in the Internet.Computer Communications, 28(7), Newman, M. E. (2005). Power laws, Pareto distributions and Zipf's law. Contemporary physics, 46(5), Wikipedia. Available at [Assessed on 12th October 2015]! 6